CET 97


 
 
 
 
 
 
 
 
 
 
                                                                                                                                                                 DOI: 10.3303/CET2297094 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Paper Received: 24 September 2022; Revised: 10 November 2022; Accepted: 23 November 2022 
Please cite this article as: Hou Y., 2022, Design of Mathematical Formula Information Retrieval System, Chemical Engineering Transactions, 
97, 559-564  DOI:10.3303/CET2297094 
 

 CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 97, 2022 

A publication of 

 
The Italian Association 

of Chemical Engineering 
Online at www.cetjournal.it 

Guest Editors: Jeng Shiun Lim, Nor Alafiza Yunus, Jiří Jaromír Klemeš 
Copyright © 2022, AIDIC Servizi S.r.l. 
ISBN 978-88-95608-96-9; ISSN 2283-9216 

Design of Mathematical Formula Information Retrieval 
System 

Yong Hou 
Bengbu University, Anhui, 233030, China 
aspnetcs@163.com 

The mathematical formula information retrieval system -MFIRS is designed and implemented, and the 
architecture of the system is discussed. A similarity indexing method based on the mathematical sub-formula of 
representation MathML is proposed. The system has the characteristics of mathematical perception. The 
mathreteval dataset was created using more than 4,500,000,000 arXiv documents and 158,106,118 
mathematical formulas, and on this dataset, The scalability of the system is verified. The front end of the system 
uses a web interface that allows users to retrieve complex queries consisting of plain text and mathematical 
formulas that can be written in TEX or MathML. When a user queries with TEX, the system is able to instantly 
convert it into a MathML tree representation and index it. The system is a mathematical formula information 
retrieval engine with mathematical perception characteristics, which can be retrieved by sub-formula similarity 
and the index of adjacent mathematical formula is realized.  

1. Introduction 
By searching in a digital library, people can find a lot of what their need. Mainstream search technology is mainly 
for plain text retrieval, text documents in the form of word bags, do not support mathematical formula processing.  
Scientific literature is full of indexes, indices, and complex mathematical formulas, even in the basic metadata, 
titles, and abstracts of papers. Research experience on Google Scholar has shown that not dealing with 
mathematical formulas in references can lead to serious retrieval problems.  
The standard for mathematical exchange between related software tools is W3C's MathML.Few people want to 
write MathML directly, and people usually prefer some kind of TEX-style compact symbol, such as LATEX or 
AMSLATEX. As a result. Mathematical retrieval system enables users to use their favorite symbols (such as 
TEX package or similar (AMS) LATEX) to query, to meet the different retrieval preferences of users, so the data 
should be converted into a unified format. Represented MathML or content MathML is used only for the output 
of software systems.  
In the process of scientific and technological literature retrieval, the unresolved mathematical retrieval problem 
becomes very prominent and arouses great interest, because the system that does not support the information 
retrieval of mathematical formula is not perfect. Therefore, The current popular mathematical retrieval systems 
are including MathDex(Chan C,2020), MathFind(Gardesten M. 2021), EgoMath(Liu H et al.,2021), Egothor(MD 
A et al.,2021), LATEXSearch(Perepu P K,2021), LeActiveMath(Shen Y et al.,2021), MathWebSearch(T V. 
Bakhteeva et al.,2021,),TUW-University of Technology(Zhai J et al.,2022,), et al.  

2. Design of the system  
The developed system divides the index content into mathematical formula index and ordinary text index when 
indexing XHTML, HTML and other documents. The indexing methods of the two types of content are different, 
among which ordinary text indexes are indexed in a conventional manner and using a traditional method. 
In Figure 1 below, the overall architecture of the system is described in detail. 

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Figure 1. The overall architecture of the system 

The index module of the realized system is mainly to normalize the input and preprocess the mathematical 
formula. 

2.1 Input Normalization 

The MathML document is normalized using the UMCL toolset to avoid the problem of mathematical formulas 
with the same semantics represented by different MathML symbols. Standardize MathML in documents through 
the UMCL toolset. 

2.2 Mathematical formula Unification 

There are three different types of unified algorithms used by the system. In order to achieve multiple common 
representations of all formulas, the unified algorithm performs a tokenization process. The system returns match 
similar to user queries, while retaining formula structures and α equations.  

2.3 Coding mathematical formula 

The mathematical formula in math format is then used with hash coding techniques and path-based coding 
techniques. Path-based coding technology splits a mathematical formula into three types of information, 
including: brother node information for subtrees, ordered path information, and no path information.  

2.4 Extract text information and mathematical formulas 

Text information is extracted from four level: Body paragraph level; Document level; Math level. The system 
realizes the overall and partial perception recognition of mathematical formulas,which appear in multiple parts 
of a document, in the following ways. 
Read all documents and use Python's regular expressions to extract, parse, and store the contents of MathML 
formulas in query documents. The specific steps are as follows. 
1).Iterate through each document and find all formulas in this document with regular expression (1). 
 
Pattern=re.compile('<m:annotation-xml id=“id\d+” encoding=“MathML-Content”>(.*?)</m:annotation-
xml>',re.S)                                                                                                                                                (1) 
 

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2).For each formula in the document, use regular expressions (2), (3), (4), (5) to extract the formula ID number, 
and get the parts of the formula. 
 
pattern = re.compile('<m:\w*\sid= "id(\d+)"',re.S)                                                                                     (2) 
pattern1 = re.compile('^(<m:\w*)?.*?',re.S)                                                                                              (3) 
pattern2 = re.compile('(.*?)</m',re.S)                                                                                                       (4) 
pattern3 = re.compile’(</m：\w*)',re.S）                                                                                                  (5) 
 
3).Construct MDG(V,E) diagrams[xx] to identify and capture variations of mathematical formulas. 
 
4).Write the extracted formula to a file and index it. 

2.5 Normalization 

The goal of normalization is to reduce MathML formula scripts with the same semantic and mathematical 
structure to only one representation, in order to solve the problem of possible inconsistencies and ambiguities 
when coding formulas in MathML. 
For example, remove the appearance elements and attributes of the presentation MathML. Individual 
appearance elements that affect the semantics of the formula, such as mathvariants, should remain in all 
possible formula elements.Unify the fence elements of MathML, trying to replace the mfenced element with the 
mrow element.Minimize the number of Mrows in MathML and remove redundant Mrow elements.Handle 
MathML's subscript/superscript elements, try to replace msubsup elements with msub and msup elements, and 
place msubs in msup.Remove the entity symbol &#x2061 that represents the function and express the function 
parameters with mrow and parentheses. 

2.6 Index math formulas 

The index of this system consists of mathematical formulas, paragraph levels index, document levels index, and 
other index information. The encoded mathematical formula and the corresponding explanatory text form the 
mathematical index. Hash coding techniques and path coding techniques are used to encode the MathML 
mathematical formulas that represent format and content type, and then generate the corresponding tables and 
place them in the index database.  

2.8 Retrieval and Ranking  

A mathematically aware, full-text-based retrieval and ranking algorithm is implemented. The algorithm is able to 
process documents that contain mathematical symbols in the presentation MathML format and filters out all 
unnecessary representation elements as well as any other MathML symbols, such as content-based MathML 
or other markup symbols. This algorithm allows the user to retrieve mathematical formulas and the text content 
of the document.  
During the retrieval phase, user input is divided into math and text. The formula is then preprocessed in the 
same way as the index stage, in addition to marketing the participle - the user may also retrieve sub-parts of the 
query formula, not just the formula as a whole.  
Similar to natural language retrieval, an algorithm can retrieve not only the entire formula, but also the individual 
sub-formulas contained in the formula.  For each formula in the input and its sub-formulas, the algorithm is able 
to create a number of different broad representations of the to allow for similarity retrieval of mathematical 
formulas. To calculate the relevance of a matching formula to a user query formula, the algorithm uses heuristic 
weighting of index words to affect the score of the matching document, thus affecting the order of the results. 
Weight formulas are assigned based on the complexity of the formula, as well as the levels in the input formula 
tree and the generalization level.  
This mathematical formula, which converts the mathematical formula in xml format into the form of compressed 
linear string, appears at the end of the preprocessing stage. 
In the retrieval stage, the final score of the document is related to the weight of the mathematical formula, and 
the calculation formula is shown in the following(6). 
 
 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠(𝑞𝑞, 𝑑𝑑) = 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑑𝑑(𝑞𝑞, 𝑑𝑑) ∙ 𝑞𝑞𝑞𝑞𝑠𝑠𝑠𝑠𝑞𝑞𝑞𝑞𝑠𝑠𝑠𝑠𝑞𝑞(𝑞𝑞) ∙ ∑ �𝑡𝑡𝑡𝑡(𝑡𝑡) ∙ 𝑖𝑖𝑑𝑑𝑡𝑡(𝑡𝑡)2 ∙ 𝑡𝑡𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔() ∙ 𝑛𝑛𝑠𝑠𝑠𝑠𝑞𝑞(𝑡𝑡, 𝑑𝑑)�𝑔𝑔∈𝑞𝑞                      (6) 
 
 
 

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3. Experimental evaluations of the system 
In this paper, a large-scale evaluation is achieved with the aid of a mathematical text library. 

3.1 Dataset 

To evaluate this system, a mathematical text library called mathreteval was built specifically. The mathreteval  
is used to evaluate the performance of this system. First, the arXiv document is converted into content-based 
XHTML format and presentation-based MathML format, Then, the resulting corpus is decompressed to a size 
of 260GB and a compressed size of 16 GB.  

3.2 Test results 

The following tests the ability of the developed system to index and retrieve relatively large real scientific 
literature repositories. The goal is to observe how the system's index file size, retrieval time, and ranking of 
retrieved documents are affected by system parameters. Check the scalability of the system. Under different 
configurations, the evaluation system has both textual and textless retrieval performance. 
The entire document set contains 158106118 formulas, and after all pre-processing, the system indexes 
2910314146 unique formulas, with an index run time of 1378 minutes (nearly 24 hours), resulting in an index 
size of approximately 88GB。  
Computing resources and experimental parameters. 512GB of RAM, 48-core Intel Xeon CPU, Ubuntu v22.04 
operating system. 
 

 

Figure 2 Scalability test results of input documents and indexed formulas  

As shown in Figure 2, the scalability of the system is approximately linear with the number of documents. This 
provides a viable response time even for billions of index sub-formulas, even if small formulas can score 
matches in most documents. We use different complex queries, such as hybrid queries, non-hybrid queries, 
high/low complexity single/multi-formula queries, and so on. to create an index, and then measure the average 
query time of the system on the mathreteval dataset, resulting in an average query time 512 milliseconds.  

 

Number of documents

N
um

be
r o

f f
or

m
ul

as

The number of input formulas

The number of indexed formulas

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Figure 3 Comparison of index consumption time of various systems 

With the increase in the number of input documents, it can be seen from Figure 3 that the index consumption 
time of each system shows an increasing trend. When the number of input documents is fixed, the index 
consumption time of the LATEXSerach system is the longest, and the index consumption time of the MFIRS 
system The shortest time. When the number of input documents reaches 1.46×106, the indexing time curves of 
the LeActiveMath, MathWebSearch and TUW systems appear to overlap. The indexing time of MFIRS shows 
an approximate linear increase. 

 

Figure 4 Comparison of index sizes of various systems 

Number of input documents

In
de

xi
ng

 ti
m

e 
(m

in
ut

es
)

LATEXSearch

LeActiveMath

MathWebSearch

TUW

MFIRS

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With an increase in the number of input documents, Figure 4 shows that the index file size of each system tends 
to increase. Among them, LeActiveMath has the largest index file, followed by MathWebsearch and 
LATEXSerach, TUW is in third place, and MFIRS has the smallest index file. When the number of input 
documents reaches about 12.9×109, the index file size of LATEXSerach, LeActiveMath and TUW is almost the 
same. 

4. Conclusions 
A method of mathematical retrieval, indexing and sorting is proposed, and it is integrated into the implemented 
system - mathematical formula information retrieval.  The feasibility of this method has been verified on the 
mathreteval data set, Verification testing of system scalability is based on this. 
Future plans are under way to use this technology in digital library projects worldwide. The system's web front 
end is fully functional and can convert mathematics into presentation MathML for full-text retrieval. The system 
is well-expanded and has the ability to be used in large digital libraries. The system can not only retrieve 
mathematical formulas of representational MathML, but also retrieve mathematical formulas of content-based 
MathML in the future. 

Acknowledgments 

This paper is supported by the following projects. Anhui Provincial Excellent Talents Training Project (Multi-
instance Learning Algorithm Research and Application (gxyq2018107); Key Project of Anhui Provincial 
Department of Education (KJ2020A0744); Bengbu College High-level Talent Start-up Project 
(BBXY2018KYQD07). 

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